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25 | <div class="section"> | |
26 | <div class="titlepage"><div><div><h3 class="title"> | |
27 | <a name="boost_multiprecision.tut.primetest"></a><a class="link" href="primetest.html" title="Primality Testing">Primality Testing</a> | |
28 | </h3></div></div></div> | |
29 | <p> | |
30 | The library implements a Miller-Rabin test for primality: | |
31 | </p> | |
32 | <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">multiprecision</span><span class="special">/</span><span class="identifier">miller_rabin</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> | |
33 | ||
34 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">expression_template_option</span> <span class="identifier">ExpressionTemplates</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Engine</span><span class="special">></span> | |
35 | <span class="keyword">bool</span> <span class="identifier">miller_rabin_test</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">number</span><span class="special"><</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">>&</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">trials</span><span class="special">,</span> <span class="identifier">Engine</span><span class="special">&</span> <span class="identifier">gen</span><span class="special">);</span> | |
36 | ||
37 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">expression_template_option</span> <span class="identifier">ExpressionTemplates</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Engine</span><span class="special">></span> | |
38 | <span class="keyword">bool</span> <span class="identifier">miller_rabin_test</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">number</span><span class="special"><</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">>&</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">trials</span><span class="special">);</span> | |
39 | </pre> | |
40 | <p> | |
41 | These functions perform a Miller-Rabin test for primality, if the result | |
42 | is <code class="computeroutput"><span class="keyword">false</span></code> then <span class="emphasis"><em>n</em></span> | |
43 | is definitely composite, while if the result is true then n is probably prime. | |
44 | The probability to declare a composite n as probable prime is at most 0.25<sup>trials</sup>. | |
45 | Note that this does not allow a statement about the probability of n being | |
46 | actually prime (for that, the prior probability would have to be known). | |
47 | The algorithm used performs some trial divisions to exclude small prime factors, | |
48 | does one Fermat test to exclude many more composites, and then uses the Miller-Rabin | |
49 | algorithm straight out of Knuth Vol 2, which recommends 25 trials for a pretty | |
50 | strong likelihood that <span class="emphasis"><em>n</em></span> is prime. | |
51 | </p> | |
52 | <p> | |
53 | The third optional argument is for a Uniform Random Number Generator from | |
54 | Boost.Random. When not provided the <code class="computeroutput"><span class="identifier">mt19937</span></code> | |
55 | generator is used. Note that when producing random primes then you should | |
56 | probably use a different random number generator to produce candidate prime | |
57 | numbers for testing, than is used internally by <code class="computeroutput"><span class="identifier">miller_rabin_test</span></code> | |
58 | for determining whether the value is prime. It also helps of course to seed | |
59 | the generators with some source of randomness. | |
60 | </p> | |
61 | <p> | |
62 | The following example searches for a prime <code class="computeroutput"><span class="identifier">p</span></code> | |
63 | for which <code class="computeroutput"><span class="special">(</span><span class="identifier">p</span><span class="special">-</span><span class="number">1</span><span class="special">)/</span><span class="number">2</span></code> is also probably prime: | |
64 | </p> | |
65 | <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">multiprecision</span><span class="special">/</span><span class="identifier">cpp_int</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> | |
66 | <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">multiprecision</span><span class="special">/</span><span class="identifier">miller_rabin</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> | |
67 | <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iostream</span><span class="special">></span> | |
68 | <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iomanip</span><span class="special">></span> | |
69 | ||
70 | <span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span> | |
71 | <span class="special">{</span> | |
72 | <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">random</span><span class="special">;</span> | |
73 | <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">;</span> | |
74 | ||
75 | <span class="keyword">typedef</span> <span class="identifier">cpp_int</span> <span class="identifier">int_type</span><span class="special">;</span> | |
76 | <span class="identifier">mt11213b</span> <span class="identifier">base_gen</span><span class="special">(</span><span class="identifier">clock</span><span class="special">());</span> | |
77 | <span class="identifier">independent_bits_engine</span><span class="special"><</span><span class="identifier">mt11213b</span><span class="special">,</span> <span class="number">256</span><span class="special">,</span> <span class="identifier">int_type</span><span class="special">></span> <span class="identifier">gen</span><span class="special">(</span><span class="identifier">base_gen</span><span class="special">);</span> | |
78 | <span class="comment">//</span> | |
79 | <span class="comment">// We must use a different generator for the tests and number generation, otherwise</span> | |
80 | <span class="comment">// we get false positives.</span> | |
81 | <span class="comment">//</span> | |
82 | <span class="identifier">mt19937</span> <span class="identifier">gen2</span><span class="special">(</span><span class="identifier">clock</span><span class="special">());</span> | |
83 | ||
84 | <span class="keyword">for</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">i</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">i</span> <span class="special"><</span> <span class="number">100000</span><span class="special">;</span> <span class="special">++</span><span class="identifier">i</span><span class="special">)</span> | |
85 | <span class="special">{</span> | |
86 | <span class="identifier">int_type</span> <span class="identifier">n</span> <span class="special">=</span> <span class="identifier">gen</span><span class="special">();</span> | |
87 | <span class="keyword">if</span><span class="special">(</span><span class="identifier">miller_rabin_test</span><span class="special">(</span><span class="identifier">n</span><span class="special">,</span> <span class="number">25</span><span class="special">,</span> <span class="identifier">gen2</span><span class="special">))</span> | |
88 | <span class="special">{</span> | |
89 | <span class="comment">// Value n is probably prime, see if (n-1)/2 is also prime:</span> | |
90 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"We have a probable prime with value: "</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">hex</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">showbase</span> <span class="special"><<</span> <span class="identifier">n</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> | |
91 | <span class="keyword">if</span><span class="special">(</span><span class="identifier">miller_rabin_test</span><span class="special">((</span><span class="identifier">n</span><span class="special">-</span><span class="number">1</span><span class="special">)/</span><span class="number">2</span><span class="special">,</span> <span class="number">25</span><span class="special">,</span> <span class="identifier">gen2</span><span class="special">))</span> | |
92 | <span class="special">{</span> | |
93 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"We have a safe prime with value: "</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">hex</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">showbase</span> <span class="special"><<</span> <span class="identifier">n</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> | |
94 | <span class="keyword">return</span> <span class="number">0</span><span class="special">;</span> | |
95 | <span class="special">}</span> | |
96 | <span class="special">}</span> | |
97 | <span class="special">}</span> | |
98 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Ooops, no safe primes were found"</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> | |
99 | <span class="keyword">return</span> <span class="number">1</span><span class="special">;</span> | |
100 | <span class="special">}</span> | |
101 | </pre> | |
102 | </div> | |
103 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
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105 | <td align="right"><div class="copyright-footer">Copyright © 2002-2013 John Maddock and Christopher Kormanyos<p> | |
106 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
107 | file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) | |
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